Switzerland, which belongs neither to the European Union nor to the European Economic Area, is not a part of the Schengen system, however, they have been cooperating with the Schengen zone and began implementing Schengen rules on December 12, 2008. Stays of Canadian nationals up to 90 days within a six-month period do not require authorisation. For longer stays, visitors have to register with the local (cantonal) migration authority at the place of your desired residence as a non-employed person. Here is a list of them:
https://www.sem.admin.ch/sem/en/home/ueberuns/kontakt/kantonale_behoerden/adressen_kantone_und.html
The corresponding EU/EFTA residence permit remains valid for a period of five years (for students, the permit remains valid for the duration of education and training or remains valid for one year and is renewed at one-year intervals). I'd suggest to travel to Switzerland while your French visa is valid and apply for a residence permit once there.